Algebro-geometric solutions of the Schlesinger systems and the Poncelet-type polygons in higher dimensions

TL;DR

This paper introduces a new algebro-geometric method for solving rank two Schlesinger systems, linking solutions of Painleve VI and Garnier systems to geometric configurations like Poncelet polygons in higher dimensions.

Contribution

It develops a novel approach to construct solutions of Schlesinger systems using meromorphic differentials on elliptic and hyperelliptic curves, generalizing classical geometric relations.

Findings

Constructed solutions relate zeros of differentials to Painleve VI and Garnier systems.

Established a connection between rational points on Jacobians and periodic billiard trajectories.

Generalized Hitchin's work linking algebraic solutions to Poncelet polygons.

Abstract

A new method to construct algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of CP^1, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painleve VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. A generalization of this differential to hyperelliptic curves is also constructed. In this case, positions of zeros of the differential provide part of a solution of the multidimensional Garnier system. The corresponding solutions of the rank two Schlesinger systems associated with elliptic and hyperelliptic curves are constructed in terms of this differential. The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has nonvariable coordinates with respect to the lattice of the Jacobian while the branch points vary. It appears that the cases where the coordinates of the point are rational correspond to periodic trajectories of the billiard ordered games associated with g confocal quadrics in (g+1)-dimensional space. This is a generalization of a situation studied by Hitchin, who related algebraic solutions of a Painleve VI equation with the Poncelet polygons.